Video Infoblog: 5, 13 and 137 are Pythagorean Primes - Numberphile

Transcript

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LAURENCE EAVES: Right.

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Well, I was born on May, which is the fifth month, and on the

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13th day of May.

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Mathematically, that's vaguely interesting because these are

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both Pythagorean primes.

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Right, OK.

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Well, if we look at 5, then we can write 5 as

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the sum of two squares.

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So 1 squared plus 2 squared is equal to 5.

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Now if we take 13--

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see if we can work that one out.

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Well, 13 is 9 plus 4, so 4 is just 2 squared.

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And 3 squared is 9.

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So that gives me--

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now, the next one, 17.

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We can do that as well, because 17 is 4 squared plus 1

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squared, which is another Pythagorean prime.

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Pythagorean--

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we've all heard of Pythagoras, who was a very famous

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mathematician.

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So why do these numbers bear his name?

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Well, if we remember our right-angle triangle.

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So we've got a right triangle there, and there's the

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hypotenuse.

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And Pythagoras tells us that if this thing is of length A,

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and this bit here is of length B, then A squared plus B

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squared equals C squared.

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That's Pythagoras's Theorem.

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Now, of course, A and B don't have to be integers.

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And C doesn't have to be an integer.

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They could be any number.

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This could be 3.2 inches.

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This could be 5.75 inches--

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whatever C is then.

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But we can play this game with integers.

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And when we do, we generate these Pythagorean primes.

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If we look at 13, then it turns out all of these numbers

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can be written as 4n plus 1.

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So 13 is equal to 4n plus 1, where n is equal to 3.

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If we take 17, then 17 is 4n plus 1 where n is equal to 4.

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So the Pythagorean primes have this.

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We generate them this way, but they've also got this

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interesting property.

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BRADY HARAN: Can I put any n in there to get a Pythagorean

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prime, or only certain n's will do it?

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LAURENCE EAVES: No, no.

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For example, if we just take the next one in this series,

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if we put n equals 5, then n equals 5 gives

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me, of course, 21.

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OK, so that's just 20 plus 1.

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And although 21 is an odd number, it's not a prime

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number because it's equal to 7 times 3.

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So 21 is not a prime.

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It's not even a prime, so it's obviously not

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a Pythagorean prime.

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So it's not all values of n that give me these, just

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special values.

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BRADY HARAN: We mentioned at the start that your birthday

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is the 13th of the 5th, and they're

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both Pythagorean primes.

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LAURENCE EAVES: Yes.

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BRADY HARAN: Can you remember at what point in your life you

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realized your birthday was two Pythagorean primes?

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LAURENCE EAVES: When you asked me to talk about numbers.

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I have a really favorite Pythagorean prime, which I'd

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like to tell you about.

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And then I just realized that 13 and 5 are

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two nice, small ones.

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BRADY HARAN: OK, tell me about your favorite one.

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LAURENCE EAVES: Well, my favorite one,

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of course, is 137.

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BRADY HARAN: For anyone that doesn't know, by the way,

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Professor Eaves is obsessed with the number 137.

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And you will see him talking about it in multiple videos.

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LAURENCE EAVES: Not really that obsessed.

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But anyway, the inverse fine-structure constant, which

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we use a lot in atomic physics and any problems to do with

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the electrons and their quantum behavior.

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The fine-structure constant alpha-- its

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inverse is almost 137.

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Not exactly 137.

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And lo and behold, 137 is a nice Pythagorean prime, which

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I think is quite nice.

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And if we work that one out, of course, we see-- how can we

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make it up?

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Well, with a little bit of thinking, we can see that 11

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squared gives me 121.

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Then if I take 4 squared, which is 16,

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that adds up to 137.

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So can we draw the Pythagorean triangle for that?

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It will look something like this.

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I've changed the scale of my units here a little bit.

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So we've got 11 there, 4 there, and then this diagonal

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is the square root of 137.

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A nice Pythagorean prime.

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BRADY HARAN: So just to be absolutely clear, the

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hypotenuse of a right-angle triangle, when the other two

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sides are integers, won't be a Pythagorean prime?

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LAURENCE EAVES: Its square root is a Pythagorean prime.

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BRADY HARAN: Its square root.

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LAURENCE EAVES: Yeah.

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MATT PARKER:That's 28.

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How many arrows do you want?

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So the next one, let's say we did 3 to the power of, to the

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power of, oh, arrow, arrow, arrow.

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Whatever you want to call this.

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JAMES GRIME: And that will give you the third perfect

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number for--

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